In practice, one discretizes the points on the Fermi surface, and solves the resulting discrete eigenvalue problem to find the dominant eigenvalues. This way, the most important features of the band structures determine the marginal BCS couplings and the dominant instabilities. Fig. REF  shows the result for the square lattice Hubbard model with second neighbor hopping \(t^{\prime }=0\)  (a) and \(t^{\prime } = -0.3t\)  (b). One sees that the dominant instability of the system in the weak-coupling limit is to a \(d\) -wave superconductor near half-filling. In a similar fashion, one can study lattice systems of other geometries [1]}, [2]}, [3]}. Among other things, Ref. [3]} found a plotting error in Ref. [2]}. Figs.1c and 1d shows results for the triangular and honeycomb lattices respectively with only nearest-neighbor hoppings. Again, near half-filling, the dominant instability is to d-wave pairing, but in this case this corresponds to a two dimensional irrep, i.e. a combination of \(d_{x^2-y^2}\)  and \(d_{xy}\)  pairing. Determining which combination of these two components is preferred in the ordered state is beyond the scope of the RG treatment, but since the effective couplings are weak, this can be addressed within the context of BCS mean-field theory. As a consequence, one expects the corresponding superconducting state to be the time-reversal symmetry breaking combination generally referred to as \(d\pm id\) .
<FIGURE>